Carney fielded several questions at Monday's press briefing about Reid's allegation, which dominated the weekend political talk shows. Republicans one-by-one have accused the Nevada Democrat of lying and suggested the Obama campaign was behind Reid's accusation. But Carney on Monday played down the notion that there was any coordination between the president's team and Reid.

"I'm not aware of the White House speaking to Senator Reid about this issue. I would simply say that you all probably know Senator Reid well, and, you know, he speaks for himself," Carney said.

Carney said that "only Senator Reid knows his source."

Carney addressed the controversy as Republicans ratcheted up their attacks on both Reid and the Obama campaign. The pushback has grown since Reid took to the Senate floor last week to say he was told Romney didn't pay taxes for 10 years -- a claim that Reid had earlier acknowledged he couldn't personally vouch for.

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After Romney first suggested Obama's team might be behind the claim, Romney surrogate John Sununu doubled down on that charge Monday in an interview with Fox News.

"The Obama campaign and President Obama are the ones that are behind this dishonesty and misrepresentation because they're trying to hide the failure of this abysmal presidency," Sununu, a former New Hampshire governor, told Fox News.

Repeating what has become something of a consensus among top Republican officials and surrogates, Sununu claimed Reid is "lying."

"Harry probably got it from talking to the mirror while he was shaving," Sununu joked, but he claimed Reid must have been "encouraged" by Obama's team.

The no-tax claim has prompted a round of name-calling unusual in its vitriol. Republican National Committee Chairman Reince Priebus called the Nevada Democrat a "dirty liar" in an interview Sunday on ABC's "This Week."

Priebus, speaking with Fox News on Monday, defended the comment.

"What else do you call somebody who goes onto the Senate floor and claims that somebody hasn't paid any taxes in 10 years, a complete lie," Priebus said. "It is what it is. I mean he's a dirty liar, and we're moving on."

But neither side was letting go entirely. Republicans have pressed Reid to "put up or shut up" -- meaning reveal his source, whom he has identified only as somebody from Romney's Bain Capital firm. Democrats, meanwhile, have used the claim to once again challenge Romney to put the matter to rest by simply putting up documents showing his extended tax history. Romney has released two years' worth of documents, but no more.

"Governor Romney can resolve this in 10 seconds. They can release the tax returns," Obama campaign adviser David Axelrod said on "Fox News Sunday."

Carney said Monday that "the issue here is one of transparency ... an important tradition that allows the American people to get a sense of a candidate's background."

Reid chief of staff David Krone told Politico earlier that Republicans are "a bunch of cowards and they're avoiding this issue."